- FindStat

Identifier

Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

[[1]] => [[1]] => 0

[[1,2]] => [[1,2]] => 0

[[1],[2]] => [[1],[2]] => 1

[[1,2,3]] => [[1,2,3]] => 0

[[1,3],[2]] => [[1,2],[3]] => 2

[[1,2],[3]] => [[1,3],[2]] => 1

[[1],[2],[3]] => [[1],[2],[3]] => 3

[[1,2,3,4]] => [[1,2,3,4]] => 0

[[1,3,4],[2]] => [[1,2,3],[4]] => 3

[[1,2,4],[3]] => [[1,2,4],[3]] => 2

[[1,2,3],[4]] => [[1,3,4],[2]] => 1

[[1,3],[2,4]] => [[1,3],[2,4]] => 4

[[1,2],[3,4]] => [[1,2],[3,4]] => 2

[[1,4],[2],[3]] => [[1,2],[3],[4]] => 5

[[1,3],[2],[4]] => [[1,3],[2],[4]] => 4

[[1,2],[3],[4]] => [[1,4],[2],[3]] => 3

[[1],[2],[3],[4]] => [[1],[2],[3],[4]] => 6

[[1,2,3,4,5]] => [[1,2,3,4,5]] => 0

[[1,3,4,5],[2]] => [[1,2,3,4],[5]] => 4

[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => 3

[[1,2,3,5],[4]] => [[1,2,4,5],[3]] => 2

[[1,2,3,4],[5]] => [[1,3,4,5],[2]] => 1

[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => 6

[[1,2,5],[3,4]] => [[1,2,3],[4,5]] => 3

[[1,3,4],[2,5]] => [[1,3,4],[2,5]] => 5

[[1,2,4],[3,5]] => [[1,3,5],[2,4]] => 4

[[1,2,3],[4,5]] => [[1,2,5],[3,4]] => 2

[[1,4,5],[2],[3]] => [[1,2,3],[4],[5]] => 7

[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => 6

[[1,2,5],[3],[4]] => [[1,2,5],[3],[4]] => 5

[[1,3,4],[2],[5]] => [[1,3,4],[2],[5]] => 5

[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => 4

[[1,2,3],[4],[5]] => [[1,4,5],[2],[3]] => 3

[[1,4],[2,5],[3]] => [[1,3],[2,4],[5]] => 8

[[1,3],[2,5],[4]] => [[1,2],[3,4],[5]] => 6

[[1,2],[3,5],[4]] => [[1,2],[3,5],[4]] => 5

[[1,3],[2,4],[5]] => [[1,4],[2,5],[3]] => 7

[[1,2],[3,4],[5]] => [[1,3],[2,5],[4]] => 4

[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => 9

[[1,4],[2],[3],[5]] => [[1,3],[2],[4],[5]] => 8

[[1,3],[2],[4],[5]] => [[1,4],[2],[3],[5]] => 7

[[1,2],[3],[4],[5]] => [[1,5],[2],[3],[4]] => 6

[[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => 10

[[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => 0

[[1,3,4,5,6],[2]] => [[1,2,3,4,5],[6]] => 5

[[1,2,4,5,6],[3]] => [[1,2,3,4,6],[5]] => 4

[[1,2,3,5,6],[4]] => [[1,2,3,5,6],[4]] => 3

[[1,2,3,4,6],[5]] => [[1,2,4,5,6],[3]] => 2

[[1,2,3,4,5],[6]] => [[1,3,4,5,6],[2]] => 1

[[1,3,5,6],[2,4]] => [[1,2,3,5],[4,6]] => 8

[[1,2,5,6],[3,4]] => [[1,2,3,4],[5,6]] => 4

[[1,3,4,6],[2,5]] => [[1,2,4,5],[3,6]] => 7

[[1,2,4,6],[3,5]] => [[1,2,4,6],[3,5]] => 6

[[1,2,3,6],[4,5]] => [[1,2,3,6],[4,5]] => 3

[[1,3,4,5],[2,6]] => [[1,3,4,5],[2,6]] => 6

[[1,2,4,5],[3,6]] => [[1,3,4,6],[2,5]] => 5

[[1,2,3,5],[4,6]] => [[1,3,5,6],[2,4]] => 4

[[1,2,3,4],[5,6]] => [[1,2,5,6],[3,4]] => 2

[[1,4,5,6],[2],[3]] => [[1,2,3,4],[5],[6]] => 9

[[1,3,5,6],[2],[4]] => [[1,2,3,5],[4],[6]] => 8

[[1,2,5,6],[3],[4]] => [[1,2,3,6],[4],[5]] => 7

[[1,3,4,6],[2],[5]] => [[1,2,4,5],[3],[6]] => 7

[[1,2,4,6],[3],[5]] => [[1,2,4,6],[3],[5]] => 6

[[1,2,3,6],[4],[5]] => [[1,2,5,6],[3],[4]] => 5

[[1,3,4,5],[2],[6]] => [[1,3,4,5],[2],[6]] => 6

[[1,2,4,5],[3],[6]] => [[1,3,4,6],[2],[5]] => 5

[[1,2,3,5],[4],[6]] => [[1,3,5,6],[2],[4]] => 4

[[1,2,3,4],[5],[6]] => [[1,4,5,6],[2],[3]] => 3

[[1,3,5],[2,4,6]] => [[1,3,5],[2,4,6]] => 9

[[1,2,5],[3,4,6]] => [[1,3,4],[2,5,6]] => 5

[[1,3,4],[2,5,6]] => [[1,2,5],[3,4,6]] => 7

[[1,2,4],[3,5,6]] => [[1,2,4],[3,5,6]] => 6

[[1,2,3],[4,5,6]] => [[1,2,3],[4,5,6]] => 3

[[1,4,6],[2,5],[3]] => [[1,2,4],[3,5],[6]] => 11

[[1,3,6],[2,5],[4]] => [[1,2,3],[4,5],[6]] => 8

[[1,2,6],[3,5],[4]] => [[1,2,3],[4,6],[5]] => 7

[[1,3,6],[2,4],[5]] => [[1,2,5],[3,6],[4]] => 10

[[1,2,6],[3,4],[5]] => [[1,2,4],[3,6],[5]] => 6

[[1,4,5],[2,6],[3]] => [[1,3,4],[2,5],[6]] => 10

[[1,3,5],[2,6],[4]] => [[1,3,5],[2,4],[6]] => 9

[[1,2,5],[3,6],[4]] => [[1,3,6],[2,4],[5]] => 8

[[1,3,4],[2,6],[5]] => [[1,2,5],[3,4],[6]] => 7

[[1,2,4],[3,6],[5]] => [[1,2,6],[3,4],[5]] => 6

[[1,2,3],[4,6],[5]] => [[1,2,6],[3,5],[4]] => 5

[[1,3,5],[2,4],[6]] => [[1,3,5],[2,6],[4]] => 9

[[1,2,5],[3,4],[6]] => [[1,3,4],[2,6],[5]] => 5

[[1,3,4],[2,5],[6]] => [[1,4,5],[2,6],[3]] => 8

[[1,2,4],[3,5],[6]] => [[1,4,6],[2,5],[3]] => 7

[[1,2,3],[4,5],[6]] => [[1,3,6],[2,5],[4]] => 4

[[1,5,6],[2],[3],[4]] => [[1,2,3],[4],[5],[6]] => 12

[[1,4,6],[2],[3],[5]] => [[1,2,4],[3],[5],[6]] => 11

[[1,3,6],[2],[4],[5]] => [[1,2,5],[3],[4],[6]] => 10

[[1,2,6],[3],[4],[5]] => [[1,2,6],[3],[4],[5]] => 9

[[1,4,5],[2],[3],[6]] => [[1,3,4],[2],[5],[6]] => 10

[[1,3,5],[2],[4],[6]] => [[1,3,5],[2],[4],[6]] => 9

[[1,2,5],[3],[4],[6]] => [[1,3,6],[2],[4],[5]] => 8

[[1,3,4],[2],[5],[6]] => [[1,4,5],[2],[3],[6]] => 8

[[1,2,4],[3],[5],[6]] => [[1,4,6],[2],[3],[5]] => 7

[[1,2,3],[4],[5],[6]] => [[1,5,6],[2],[3],[4]] => 6

[[1,4],[2,5],[3,6]] => [[1,4],[2,5],[3,6]] => 12

[[1,3],[2,5],[4,6]] => [[1,3],[2,5],[4,6]] => 9

>>> Load all 1884 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

Map

Schützenberger involution

Description

Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.